Shut
up the words, and seal the book, even to the end of time. Many shall run to
and fro, and knowledge shall be increased.

Daniel
12:4

I
gave up my heart to know wisdom, and to know maddness and folly. I perceived
also that this is vexation of spirit. For in much wisdom is much grief, and
he that increaseth knowledge increaseth sorrow.

Ecclesiastes,
1:17

When discussing the difference between physics and
mathematics, people sometimes acknowledge that there are paradigm shifts in
mathematics, but they maintain that these shifts do not invalidate previous
results, whereas in physics (they claim) a paradigm shift does
invalidate previous results. As an example, they suggest that the ancient belief
that everything is composed of earth, wind, fire and water has been falsified
by modern physics, whereas general relativity has not falsified the proof of
the Pythagorean theorem found in Euclid.

Let's take these one at a time. First, as a classification
scheme for the possible states of matter, the categories of solid, gas,
liquid, and plasma (earth, wind, water, fire) are not too bad. Then again,
it's not clear that it's even possible to "invalidate" a
classification scheme. Can we invalidate the classification of Europe and Asia
as separate continents? People have reasons for classifying things as they
do. We may decide to classify things differently someday (although those four
categories have been remarkably enduring), but that wouldn't really
"invalidate" other schemes.

Second, does general relativity prove the ancient Greek
geometers were "wrong"? There's a sense in which it actually does,
insofar as they were engaged in an activity that today would be called
physics, i.e., they were trying to find out facts about spatial relations. For
example, we know from a letter written by Archimedes to a friend that he
actually discovered many of his marvelous theorems by cutting out curved
shapes (sections of parabolas and so on), and then physically weighing them
to compare their areas, and dunking spheres and cylinders in water and
measuring their displacements to compare their volumes. Admittedly he then
went on to devise synthetic proofs of his theorems, but he clearly regarded
his results as descriptive of actual spatial relations. In this he was
mistaken.

How does this bear on the issue at hand? In one sense it
supports the idea that physical theories have been invalidated whereas math
theories have not, if we accept that Euclidean geometry has been invalidated
as a physical theory but not as a mathematical theory. However, in two ways,
one technical and one philosophical, it also shows how mathematical ideas
have been invalidated and rejected.

The technical point is that if we remove the empirical
justification for Euclid's geometry (for example), and try to evaluate it as
a purely abstract mathematical creation, it doesn't stand up all that well to
modern scrutiny. In a way this is inevitable because Euclid wasn't trying
to create an arbitrary axiomatic system in the modern sense. Nevertheless, if
mathematicians want to claim the Elements as a work of pure math they can't
escape the fact that it fails strictly on those terms. For example, there are
no axioms of "betweeness" or "continuity" or many other
things that would be necessary for a coherent axiomatic system. To compensate
for these deficiencies, Euclid relies heavily on physical intuition and
visual evidence. So the direct answer to the question is yes, the modern judgment
of Euclid's proof of Pythagoras's theorem is that it is incorrect, i.e., it
does not follow by strict deductive reasoning from the axioms and postulates
and common notions as presented.

The second point is philosophical. People like Euclid and
Archimedes considered themselves to be mathematicians and believed what they
were doing - finding out the true ideal forms - was mathematics. As a
mathematical idea this view has been largely invalidated. Today most
mathematicians believe that math is the business of working out the
implications of a set of premises - any set of premises that strikes
their fancy. In this sense modern mathematics has evolved a new understanding
of what math is, and has largely rejected and discarded the view of
mathematics held by most ancient (and some not so ancient) mathematicians.

In the popular imagination, Isaac Newton (one of the
founders of modern physics) is often regarded as espousing a somewhat naive
mechanistic and deterministic world-view, but in fairness it should be said
that Newton was (in public) admirably circumspect about the underlying
structure of reality. He even claimed that "I make no hypothesis". In
fact he was severely criticized on precisely these grounds, i.e., he insisted
on simply describing things, and declined to offer explanations of things.
He knew the difference between a scientific theory and an interpretation. When
people talk about scientific theories being overthrown, they usually mean an interpretation
has been replaced.

The classical case is Ptolemy's astronomy, which was used
to describe and predict events with acceptable accuracy for centuries. It was
never disproved - it still works today as well as it ever did. It was simply
replaced by a different interpretation that proved to be more comprehensive,
elegant, and powerful. Of course, what people have in mind when they say Ptolemy’s
theory was rejected is the change in interpretation. Compare this with the
history of mathematics. As an example, consider the old Theory of Equations
which was studied and developed for centuries. Eventually it was superceded
by Galois Theory and abstract algebra, and the old theory was largely
abandoned. This doesn't mean the old theory was wrong - it still works as
well as it ever did. It was simply replaced by a more comprehensive, elegant,
and powerful theory. Moreover, I would argue that the advent of abstract
algebra, with its non-commutative multiplications and so on, represented a
real change in the interpretation of the subject matter. The old "permanence
of forms" was overthrown. Thus, even if we go back and look at an old
book on the Theory of Equations, we will see it in a different light and
attribute to it a somewhat different meaning than the author had in mind. We
now think of algebras (plural),
rather than conceiving of One True Algebra located eternally at the center of
the universe. All the observables may be the same, but our idea of "what
is really going on" has changed.

In short, I think mathematical theories and
interpretations have evolved and changed throughout history much like the
theories and interpretations in other branches of knowledge. The view that mathematical
knowledge is uniquely enduring is based on an under-estimation of the extent
to which past mathematical ideas have been displaced, and an overestimation
of how much knowledge in other areas has actually been falsified (as opposed
to re-interpreted). Of course, the classical counter-argument to the claim
that mathematical knowledge alone among all branches of knowledge is
cumulative, is to point out that if this were really the case we would expect
to have no more knowledge of physics (for example) today than we did at the
beginning of recorded history, and we would expect the majority of our
knowledge to be mathematical, since even if its rate of accumulation is slow,
the fact that it is cumulative should eventually make it overwhelm every
other branch of knowledge. Does our experience support these expectations? I
would say no. Since the time of Archimedes, which branch of knowledge has
seen more cumulative progress, mathematics or physics?